Section 4.4 Logarithmic Functions
Definition:
2) A logarithmis
merely a name for a certain exponent!
Important Result…1) The log function
and the exponential functions are inverses of each other!
xbyx yb log
1. Special Logarithms - base 10 and e
ya axxy log
2. Changing from log to exponential form
3log 1) 2 x
2log 2) 3 x
)log(2 )3 x
xln2 4) y
a axxy log
x416 5)
x5125 6)
3. Changing from exponential to log form
x10001 7)
xe51 8)
ya axxy log
1log 1) 2
16log 3) 2/1
4. Evaluating Logarithms
81
1log 2) 9
10log 4)
4e ln 5)y
a axxy log
ya axxy log
5. Special Properties
Inverse Functionsxbxf )(
Switch and solve:1) Replace f(x) with y:2) Interchange x and y:3) Solve for y:4) Replace y with
xxf blog)(1
6. Log is the Inverse of xbxf )(
)(1 xf
Inverse Property of Logarithms
xb
xbx
xb
b
log
log
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
xxf 2)(
Sketch the inverse of
reflecting graph of
over
)(log)( 21 xxf
Domain:Range:Key Points:Asymptotes:
7.Graphing the inverse of xxf 2)( xxf 2)(
xy 2
xy
Graph directlyxxf 5log)( a) Key Points
b) Domain
c) Range
d) Asymptotes
xxf 2/1log)(
a) Key Points
b) Domain
c) Range
d) Asymptotes
is the inverse of
8 Two Methods for Graphing Log Functions
Method 1: Directly
Method 2: Use Inverse
1)2(log)( 2 xxf
a) Key Points of parent
b) Transformations
c) Domain
d)Range
e) Asymptotes
Method 1: Determine the graph of a log function, using transformations of the parent function.
1)2(log)( 2 xxf
1) Find 2) Graph3) Reflect over y = x to graph
)(1 xf
)(1 xf
)(xf
Method 2: Determine the graph of a log function by graphing its inverse function and reflect over y=x
9. Domain of a logarithmic function
Determine the domain for these functions.
1ln)(x
xxf
12log)( 4 xxf
10. Change-of-Base Formula
log lnlog
log lnaM M
Ma a
Example.
Find an approximation for )5(log2
11. Solving: Log = Constant
Logarithmic Equations
Solve for variable inside the log expression.
15)82(log 4 x
Use the definition: xC blog xbC
12. Solving: Exponential = Constant
Solve for variable in the exponent
Use the definition: Cx blogCb x
723 1 x
11. Solving: Log = Log
If then M = N NM aa loglog
)12(loglog 1) 33 xx
Solve for variable inside log on each side.
When solving log functions, we must
check that a solution lies in the
domain!
Summary: Inverse Properties of Logarithmic and Exponential Functions
The Logarithmic and Exponential Functions are inverses of each other.
Example of the relationship: Let xxfxg x2log)( and 2)(
Inverse Property of xxfbxg bx log)( and )(
xbxgfbxfg xb
xb log))(( andx ))(( log
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
xxf 3)(
Exponential functions and log functions are inverse functions
of each other.
xxf 31 log)(
Domain:Range:Key Points:Asymptotes:
Graphing Logarithmic Functions